What kinds of systems have so far failed to be modeled via supervised artificial network training?
I haven't seen any dataset where some standard models worked and neural networks utterly failed.

For columnar data (e.g. Excel files / database dumps / CSV files) which contain structured data usually tree-based models like random forests and gradient boosting work better, but neural networks are also usually way better than random.

If you demand other things, e.g. explanations for the decision then Bayesian models might give you an easier time. Or for baselines/simple implementations linear models. Or for real time applications...